The theory of probabilistically checkable proofs [1]}, [2]} was instrumental in proving a host of \(\text{NP}\) -hardness of approximation results. Though this theory was very successful in settling the time-complexity for approximation problems in NP, there are inherent limitations to extend these techniques to problems in P. Towards this, there has been recent progress for establishing hardness of approximation results for problems in P [3]}, [4]}, [5]}. Our paper makes further progress in this direction.
